where \(o_i\)  is the POVM measurement outcome (single shot). Hence, it is a random variable that depends on \(\rho \)  and \(F_i\) .
This setup is known as single-copy independent measurements in the quantum state tomography literature [1]}, [2]}.
Without loss, we can further restrict our attention to POVM measurements comprised of rank-one projectors. Such measurements always reveal more information and we write
\(F_i = \lbrace w_{i o_i} 2^n \vert \psi _{i o_i} \rangle \! \langle \psi _{i o_i} \vert \rbrace _j\) .
The classical ML model then uses the classical measurement outcomes \(o_1, \ldots , o_N\)  to learn a function \(f(x)\)  such that the following prediction error bound holds with high probability:
\(\max _{x \in \lbrace I, X, Y, Z\rbrace ^n} \left| f(x) - \operatorname{tr}(Z_1 \operatorname{{\mathcal {E}}}_\rho (\vert x \rangle \! \langle x \vert )) \right| = \max _{x \in \lbrace I, X, Y, Z\rbrace ^n} \left| f(x) - \operatorname{tr}(P_x \rho ) \right| < \frac{1}{2}.\) 
